Notes on the Cauchy Problem for Backward Stochastic Partial Differential Equations

Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W_2^n$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275–293 (1992)]. As an application, a comparison theorem is obtained.